Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs
By F. Bonetto, Joel Lebowitz, Jani Lukkarinen. 2003
165 citations recorded by the source corpus.
We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the “exterior” left and right heat baths are at specified values TLand TR, respectively, while the temperatures of the “interior” baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fourier's law with a finite positive thermal conductivity which can also be computed using the Green–Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich, and Visscher in 1970 who first studied this model. In the other limit, d>>1, the stationary infinite volume heat conductivity behaves as (ldd)−1where ldis the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit.
Subfields: Materials Chemistry, Mechanics of Materials, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/a625a82238d9b65f111b5e07c74d81e3012cf5d8
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In Materials Chemistry
In Mechanics of Materials
In Statistical and Nonlinear Physics
Citations in the corpus
- Heat transport in low-dimensional systems (2008, methodology): “The ordered harmonic lattice with self-consistent reserviors was solved exactly by Bonetto et al. [36], in arbitrary dimensions, who proved local equilibration and validity of Fourier’s law, and obtained an expression for the thermal conductivity of the system.”
- Fourier’s Law for a Microscopic Model of Heat Conduction (2005, background): “configuration space and by (ωs)s0 the Markov process with generator ( 3 ).”
- Non-equilibrium steady states: fluctuations and large deviations of the density and of the current (2007, background): “Beyond trying to know the steady state measure P (C), which can be done only for a very few examples [7]–[13], one might wish to determine a number of properties of non-equilibrium steady states like the temperature or energy profiles [14, 15], the average flow of energy through the system [ 16 ]–[2”